Moment Infinitely Divisible Weighted Shifts

Abstract

We say that a weighted shift \(W_\alpha \) with (positive) weight sequence \(\alpha : \alpha _0, \alpha _1, \ldots \) is moment infinitely divisible (MID) if, for every \(t > 0\), the shift with weight sequence \(\alpha ^t: \alpha _0^t, \alpha _1^t, \ldots \) is subnormal. Assume that \(W_{\alpha }\) is a contraction, i.e., \(0 < \alpha _i \le 1\) for all \(i \ge 0\). We show that such a shift \(W_\alpha \) is MID if and only if the sequence \(\alpha \) is log completely alternating. This enables the recapture or improvement of some previous results proved rather differently. We derive in particular new conditions sufficient for subnormality of a weighted shift, and each example contains implicitly an example or family of infinitely divisible Hankel matrices, many of which appear to be new.

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Acknowledgements

The authors wish to express their gratitude to an anonymous referee for detecting an omission in the original statement of Theorem 3.1. The authors are also indebted to another anonymous referee for a detailed reading of the paper, and for several helpful suggestions which led to improvements in the presentation. Some of the proofs in this paper were obtained using calculations with the software tool Mathematica [22].

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Correspondence to Raúl E. Curto.

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The first named author was partially supported by Labex CEMPI (ANR-11-LABX-0007-01). The second named author was partially supported by NSF Grant DMS-1302666. The third named author was partially supported by Labex CEMPI (ANR-11-LABX-0007-01).

Communicated by Joseph Ball.

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Benhida, C., Curto, R.E. & Exner, G.R. Moment Infinitely Divisible Weighted Shifts. Complex Anal. Oper. Theory 13, 241–255 (2019). https://doi.org/10.1007/s11785-018-0771-z

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Keywords

  • Weighted shift
  • Subnormal
  • Completely monotone sequence
  • Completely alternating sequence
  • Moment infinitely divisible

Mathematics Subject Classification

  • Primary 47B20
  • 47B37